Lighthill's Acoustic Analogy
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∂ 2
ρ
∂t2 − a2s ∂
2
ρ
∂x2i= ∂
2
τ ij∂xi∂x j
ρ τ ij as
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∂ρ
∂t +
∂ρvi∂xi
= 0
ρ v i − th v
ρ∂vi∂t
+ ρv j∂vi∂x j
= −∂p
∂xi−
∂τ ij∂x j
= ∂
∂x j( pδ ij + τ ij)
p τ ij δ
[τ ]
τ ij = −µ
∂vi∂x j
+ ∂v j∂xi
+
2
3µδ ij
∂vk∂xk
µ v p p − p0
ρ∂vi∂t
+ vi∂ρi∂t
+ ρv j∂vi∂x j
+ vi∂ρv j∂x j
= ∂
∂x j(( p − p0)δ ij − τ ij)
τ I
τ I ij = ρviv j + ( p
− p0)δ ij − τ ij
∂ρvi∂t
=∂τ I ij∂x j
τ uij = ( p − p0)δ ij
∂ρvi
∂t +
∂
∂t( p − p0) = 0
p = p − p0
ρ = ρ − ρ0
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p
ρ = c2
p0 ρ0 c
p − p0 c2ρ ∂/∂xi
∂
∂t
∂
∂xi(ρvi) +
∂ 2
∂x2i(c2ρ) = 0
∂ 2ρ
∂t2 −
∂ 2
∂x2i(c2ρ) =
1
c2∂ 2
∂t2 −
∂ 2
∂x2i
(c2ρ) = 0
ρ = ρ − ρ0 = 0
Lij = τ I ij − τ
0
ij
Lij = ρviv j + (( p − p0) − c2(ρ
− ρ0))δ ij − τ ij
Lij
∂ρvi∂t
+ ∂τ ij∂xi
=∂ (τ I ij − τ
0ij)
∂xi
∂ρvi∂t
+ ∂
∂xi(c2(ρ − ρ0)) = −
∂Lij∂xi
ρvi
1
c2∂ 2
∂t2 −
∂ 2
∂x2i
(c2(ρ − ρ0)) = −
∂ 2Lij∂xi∂x j
vi
ρviv j eij
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(( p − p0) − c2(ρ − ρ0))
Lij ≈ ρviv j
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ρ p
LoΠ = Γ
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Lo ≡ DoDt
Do2
Dt2 −
∂ ∂xi
c2 ∂ ∂xi
+ 2 ∂U
∂xi∂
∂x1c2 ∂
∂xiDoDt
= ∂ ∂t
+ U ∂ ∂x1
Π ≡ 1κ
ln p p0
c2 = κRT o
∂Q
∂t +
∂ E
∂x +
∂ F
∂y +
∂ G
∂z =
∂E υ∂x
+ ∂ F υ
∂y +
∂Gυ∂z
Q =
ρρuρυρωρE
E =
ρuρu2 + p
ρuυρuω
(ρE + p)u
F =
ρυρυu
ρυ2 + pρυω
(ρE + p)υ
G =
ρωρωuρωυ
ρω2 + p(ρE + p)ω
E υ =
0τ xxτ xyτ xz
uτ xx
+ υτ xy
+ ωτ xz− q
x
F =
0τ yxτ yyτ yz
uτ yx + υτ yy + ωτ yz −
q y
G =
0τ zxτ zyτ zz
uτ zx + υτ zy + ωτ zz −
q z
q x q y q z
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p2 = p12γM 2s − γ +
1
γ + 1
ρ2 = ρ1
(γ + 1)M 2s
(γ − 1)M 2s + 2
u2 = cs
2(M 2s − 1)
(γ + 1)M 2s
cs ρ1 p1 a0
ρ2 p2 u2
M s = cs/a0
x > a0t t = 0
ρ(x, 0) = f (x) ∂ρ∂t
(x, 0) = g(x)
f = ρ1 g = 0
ρ(x, t) =
ρ1, x > cst
ρ2, a0t < x < cst
ρ2 = ρ1 + 1
a20− c2s
2γp1(M
2s − 1)
γ + 1 −
2a20 p1(M
2s − 1)
(γ − 1)M 2s + 2 −
4ρ1(M 2s − 1)
(γ − 1)M 2s + 2
1
(γ − 1)M 2s
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https://www.dropbox.com/s/cj8ips08dnihz62/Flow%20Induced%20Noise%20Simulation.pdf/https://www.dropbox.com/s/cj8ips08dnihz62/Flow%20Induced%20Noise%20Simulation.pdf/
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http://www.cfd.tu-berlin.de/index.php?sec=research&subsec=acoustics&subsubsec=overview&lang=english/http://www.cfd.tu-berlin.de/index.php?sec=research&subsec=acoustics&subsubsec=overview&lang=english/
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